PERIODIC MOTION 319 where & is a positive constant which depends upon the physical properties of the wire.* The negative sign indicates the fact that the torque and the angular displacement are oppositely directed. Substituting this value of G in the torque equation we have /f = -ke, (xvii) *--«• k where c2= -• But these are the typical forms of the equation of simple harmonic motion; therefore (XVIII) is the expression for the period. It will be observed that the motion is strictly harmonic; consequently there is no correction for finite amplitudes. 247. Application to the Determination of Moment of Inertia. — Let P be the period of the torsion pendulum and Pr its period after the body whose moment of inertia is desired is fastened to the bob of the pendulum. Further let I be the moment of inertia of the bob about the suspension wire as an axis and T the moment of inertia of the body. Then we have P=2 and P' = 2 P'2_ p2 Therefore I'= p2 I i A « I' and k = 4 IT* ~-T^—; Hence if I is known both I' and k may be determined experimentally. * Page 178.